Approximation methods for hybrid diffusion systems with state-dependent switching processes: numerical algorithms and existence and uniqueness of solutions

G. Yin, X. Mao, C. Yuan, D. Cao

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20 Citations (Scopus)

Abstract

By focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. Different from the existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing stepsize algorithms together with their weak convergence are given. Numerical experiments are also provided for demonstration.
LanguageEnglish
Pages2335-2352
Number of pages18
JournalSIAM Journal on Mathematical Analysis
Volume41
Issue number6
DOIs
Publication statusPublished - 13 Jan 2010

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Existence and Uniqueness of Solutions
Approximation Methods
Numerical Algorithms
Weak Convergence
Dependent
Picard Iteration
Martingale Problem
Euler Scheme
Discrete Event
Numerical Procedure
Strong Convergence
Convergence Results
Numerical Scheme
Iterative Algorithm
Stochastic Equations
Existence and Uniqueness
Differential equations
Demonstrations
Numerical Experiment
Differential equation

Keywords

  • switching diffusion
  • numerical algorithm
  • convergence
  • strong solution
  • differential equations

Cite this

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AB - By focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state dependent switching process presents added difficulties in analyzing the numerical procedures. Weak convergence of the algorithms is established by a martingale problem formulation first. This weak convergence result is then used as a bridge to obtain strong convergence. In this process, the existence and uniqueness of the solution of the switching diffusions with continuous-state-dependent switching are obtained. Different from the existing results of solutions of stochastic differential equations in which the Picard iterations are utilized, Euler's numerical schemes are considered here. Moreover, decreasing stepsize algorithms together with their weak convergence are given. Numerical experiments are also provided for demonstration.

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